7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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12. According to Bortkiewicz’ terminology, the latter corresponds to the law of smallnumbers (to the terms of a binomial of an increasing degree with the probability tending tozero.13. {On Fechner see Sheynin (2004).}14. As compared with social statistics, biology also partly enjoys the same benefit.15. {It was Ellis (1850, p. 57) rather than Poincaré: Mere ignorance is no grounds for anyinference whatever. … ex nihilo nihil.References1. Anonymous (1962), B.S. Yastremsky. An obituary. Vestnik Statistiki, No. 12, pp. 77 –78. (R)2. Bernstein, S.N. (1926), Sur les courbes de distribution des probabilités. Math. Z., Bd. 24,pp. 199 – 211.3. --- (1946), (Theory of Probability). M., 4th edition.4. Bertrand, J. (1888), Calcul des probabilités. Paris.5. Borel, E. (1914), Le hasard. Paris.6. Ellis, R.L. (1850), Remarks on the alleged proof of the method of least squares. Inauthor’s Math. and Other Writings. Cambridge, 1863, pp. 53 – 61.7. Liapunov, A.M. (1901), Answer to P.A. Nekrasov. Zapiski Kharkov Univ., vol. 3, pp. 51– 63. Translated in Nekrasov, P.A. (2004), Theory of Probability. Berlin.8. Marbe, K. (1899), Naturphilosophische Untersuchungen zur Wahrscheinlichkeitsrechnung.Leipzig.9. Poincaré, H. (1912), Calcul des probabilités. Paris. First edition, 1896.10. Sheynin, O. (1990, in Russian), Chuprov. Göttingen, 1996.11. --- (1994), Bertrand’s work on probability. Arch. Hist. Ex. Sci., vol. 48, pp. 155 – 199.12. --- (2004), Fechner as a statistician. Brit. J. Math. Stat. Psychology, vol. 57, pp. 53 – 72.13. Yastremsky, B.S. (1964), (Sel. Works). M.2. A.Ya. Khinchin. The Theory of Probability. In 15 . (Science in the Soviet Union during 15 Years.Mathematics). Editors, P.S.Aleksandrov et al. Moscow – Leningrad, 1932, pp. 165 – 169[Introduction] The theory of probability is nearly the only branch of mathematics where,as also acknowledged abroad, even the pre-revolutionary Russian science from the time ofChebyshev had been occupying a leading position. The responsibility for the maintenanceand strengthening of this leading part naturally lay on the Soviet mathematicians and becameeven greater and more honorable since, during the last 10 – 15 years, the European scientificthought in the sphere of probability theory (Italy is here on the first place, then come theScandinavian countries, Germany and France) considerably advanced from its infantile stageand rapidly attained the level established in Russia by the contribution of Chebyshev,Markov and Liapunov. The foremost European schools at least qualitatively even exceededthat level by coming out at once on the wide road provided by the modern methods ofmathematical analysis and free from the touch of provincialism from which (in spite of thegreatness of its separate findings) our pre-revolutionary scientific school neverthelesssuffered.It seems that now, after 15 years of work 1 , we, Soviet mathematicians, may state that wehave with credit accomplished the goal that had historically fallen to our lot. Notwithstandingthe already mentioned very considerable advances of the West-European scientific thought,today also the Soviet probability theory is occupying the first place if not in accord with thenumber of publications, but in any case by their basic role and scientific level. The present-
day portrait of this discipline has no features, nor has its workshop a single range of problemswhose origin was not initiated by Soviet mathematics. And the level of our science is bestdescribed by noting that European mathematicians are until now discovering laws known to,and published by us already five – ten years ago; and that they (sometimes includingscientists of the very first rank) publish results which were long ago introduced into ourlectures and seminars and which we have never made public only because of consideringthem shallow and immature if not altogether trivial.It is interesting to note that after the October upheaval 2 our probability theory underwent aconsiderable geographical shift. In former times, it was chiefly cultivated in Leningrad {inPetersburg/Petrograd} but during these 15 years Leningrad {Petrograd until 1924} did notoffer any considerable achievements. The main school was created in Moscow (Khinchin,Kolmogorov, Glivenko, Smirnov, Slutsky) and very substantial findings were made inKharkov (Bernstein) and Tashkent (Romanovsky).Before proceeding to our brief essay on those lines of development of the theory ofprobability which we consider most important, we ought to warn our readers that it is not ourgoal to provide even an incomplete list of Soviet accomplishments in this sphere; we aim attracing the main issues which occupied our scientists and at indicating the essence which wecontributed to the international development of our science, but we do not claim anycomprehensiveness here. Then, we ought to add a reservation to the effect that we restrict ourdescription to those lines of investigation which have purely theoretical importance, and weleave completely aside all research in the field of practical statistics and other applications ofprobability in which Soviet science may also take pride. We consider the following issues ashaving been basic for the development of the Soviet theory of probability.1. Investigations connected with the so-called limit theorem of the theory of probability,i.e., with the justification of the Gauss law as the limiting distribution for normed sums ofrandom variables. Formerly, almost exclusively considered were the one-dimensional caseand series of mutually independent random variables. Nowadays we are able to extend thelimit theorem, on the one hand, to the many-dimensional case, and, on the other hand, toseries of mutually dependent variables. Soviet mathematicians and Bernstein in the first placeboth initiated these generalizations and achieved the most important discoveries {here}.Bernstein accomplished fundamental results in both directions; Kolmogorov arrived at someextensions; Khinchin justified normal correlation by the direct Lindeberg method. Thissphere of issues met with a most lively response in the European literature and continues tobe developed both at home and abroad.2. For a long time now, the theory of functions of a real variable, that was worked out atthe beginning of our {of the 20 th } century, compelled mathematicians to feel a number ofdeep similarities connecting the stochastic concepts and methods with the main notions of themetric theory of sets and functions. The appropriate process of modernizing the methodologyof probability theory, its notation and terminology, is one of the most important aspects ofthe modern issues of our scientific domain. This process is far from being completed, but wealready feel with complete definiteness how much it is offering to probability theory, – howmuch its development fosters the unity and the clarity of the scientific method, the harmonyand the visibility of the scientific building itself. With respect to their level, even the mostpowerful stochastic schools (for example, the Italian school) insufficiently mastering themethods of the function theory become noticeably lower than those (the French, the Moscowschool) where these methods are entrenched. It is therefore quite natural that Moscowmathematicians, most of whom had developed in the Luzin school, have marched in the frontline of this movement. Slutsky should be named here in the first instance, then Glivenko andKolmogorov. In particular, the last-mentioned had revealed, with exhausting depth and
Page 4 and 5: [3] I bear in mind the well-known p
Page 6 and 7: successes of physical statistics. B
Page 8 and 9: classes of independent facts whose
Page 10 and 11: distribution is a corollary of the
Page 12 and 13: examine in the first place the curv
Page 16 and 17: generality, the similarities taking
Page 18 and 19: one on another, as well as the corr
Page 20 and 21: is inapplicable because the right s
Page 22 and 23: Instead, Slutsky introduced new not
Page 24 and 25: abandoned in August 1936, but it is
Page 26 and 27: last decades, mathematicians more o
Page 28 and 29: charged with making the leading ple
Page 30 and 31: motion and a number of others) are
Page 32 and 33: phenomena. It is self-evident that
Page 34 and 35: Such new demands were formulated in
Page 36 and 37: The addition of independent random
Page 38 and 39: automatic lathes, etc. Here, the ma
Page 40 and 41: 11. Kolmogorov, A.N. Grundbegriffe
Page 42 and 43: period 1 and remained, until the ap
Page 44 and 45: of the analytical tool rather than
Page 46 and 47: with probability approaching unity,
Page 48 and 49: logic. The ensuing vagueness in his
Page 50 and 51: 2. Gnedenko, B.V. (1949), On Lobach
Page 52 and 53: will be sufficient, although not ne
Page 54 and 55: nlimk = 1P(| k (n) - m k (n) | > H
Page 56 and 57: favorite classical issue as the gam
Page 58 and 59: and some quite definite (not depend
Page 60 and 61: influenced by a construction that a
Page 62 and 63: P ij (1) = p ij (1) , P ij (t) =kP
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4.2d. Bebutov [1; 2] as well as Kry
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are yet no limit theorems correspon
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described, from the viewpoint that
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conditional variance and determined
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Romanovsky [45] and Kolmogorov [46]
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Let S be the general population wit
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Part 1. Russian/Soviet AuthorsAmbar
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2. On necessary and sufficient cond
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Gnedenko, B.V., Groshev, A.V. 1. On
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52. ( (Mathematical Principl
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Kozuliaev, P.A. 1. Sur la répartit
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Obukhov, A.M. 1. Normal correlation
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30. Généralisations d’un théor
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22. Alcune applicazioni dei coeffic
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10. A.N. Kolmogorov. The Theory of
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Kuznetsov, Stratonovich & Tikhonov
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In the homogeneous case H s t = H t
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to such a generalization. He only s
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In the particular case of a charact
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as it is usual for the modern theor
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1. {The second reference to Pugache
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Smirnov, Romanovsky and others made
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determined the precise asymptotic c
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for finite values of N, M and R 2 .
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Mikhalevich’s findings by far exc
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Uch. Zap. = Uchenye ZapiskiUkr = Uk
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Khinchin, A.Ya. 43. Math. Ann. 101,
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7. DAN 115, 1957, 49 - 52.Pinsker,
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Anderson, T.W., Darling, D.A. (1952
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Statistical problems in radio engin
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observations for its power with reg
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securing against mistakes (A.N. Kry
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of the others, then its distributio
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In Kiev, in the 1930s, N.M. Krylov
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7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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